generalization of the kermack-mckendrick sir model to a ...generalization of the kermack-mckendrick...

Math. Model. Nat. Phenom.Vol. 4, No. 2, 2009, pp. 92-118

DOI: 10.1051/mmnp/20094205

Generalization of the Kermack-McKendrick SIR Modelto a Patchy Environment for a Disease with Latency

J. Li∗ and X. Zou†

Department of Applied Mathematics University of Western OntarioLondon, Ontario, Canada N6A 5B7

Abstract. In this paper, with the assumptions that an infectious disease has a fixed latent pe-riod in a population and the latent individuals of the population may disperse, we reformulate anSIR model for the population living in two patches (cities, towns, or countries etc.), which is ageneralization of the classic Kermack-McKendrick SIR model. The model is given by a systemof delay differential equations with a fixed delay accounting for the latency and non-local termscaused by the mobility of the individuals during the latent period. We analytically show that themodel preserves some properties that the classic Kermack-McKendrick SIR model possesses: thedisease always dies out, leaving a certain portion of the susceptible population untouched (calledfinal sizes). Although we can not determine the two final sizes, we are able to show that the ratioof the final sizes in the two patches is totally determined by the ratio of the dispersion rates ofthe susceptible individuals between the two patches. We also explore numerically the patterns bywhich the disease dies out, and find that the new model may have very rich patterns for the diseaseto die out. In particular, it allows multiple outbreaks of the disease before it goes to extinction,strongly contrasting to the classic Kermack-McKendrick SIR model.

Key words: infectious disease, SIR model, latent period, patch, non-local infection, dispersion,multiple outbreaksAMS subject classification: 34K18, 34K20, 34D 23, 37N25, 92D30

∗Current address: Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON K1N 6N5,Canada.†Corresponding author. E-mail: [email protected]

92

Article available at http://www.mmnp-journal.org or http://dx.doi.org/10.1051/mmnp/2009007

http://www.mmnp-journal.org

http://dx.doi.org/10.1051/mmnp/2009007

J. Li and X. Zou. A generalization of the Kermack-McKendrick SIR model on patches

1. IntroductionKermak-McKendrick model [7] is one of the earliest triumphs in mathematical epidemiology [4].This simple model is formulated for a population being divided into three disjoint sub-populationsor compartments—susceptible class S, infective class I and removed class R, and is given by thefollowing system of ordinary differential equations

dS

dt= −λSI,

dI

dt= λSI − σI,

(1.1)

with the R class being determined by dRdt

= σI . Here the removed class could include the recoveredindividuals with permanent immunity, as well as the individuals who die of the disease in the casethat the disease causes deaths. The constant σ > 0 is called the removal rate and the constantλ > 0 is called the infection rate. Given S(0) = S0 > 0 and I(0) = I0 > 0, the analysis of (1.1)has shown (see, e.g., [5, 11]) that

(i) when R0 := λS0/σ < 1, there is no outbreak of the disease in the sense that the population ofthe infectious class I(t) decreases monotonically to 0;

(ii) when R0 := λS0/σ > 1, there will be a single outbreak of the disease in the sense that I(t)firstly increases monotonically to a maximum value, and after that, I(t) decreases monoton-ically to 0;

(iii) in either of the above two situations, the disease eventually dies out of the population, leavingpart of the population, denoted by S∞, untouched by the disease.

The quantityR0 defined above is referred to as the basic reproduction number of the SIR model(1.1), accounting for the average number of new infections that a single infectious individual cancause during the infection life time (see, e.g., [1, 5]). The untouched part S∞ of the susceptibleclass is often referred to as the final size of the SIR model (1.1), and is determined implicitly bythe equation

I0 + S0 − σ

λln S0 = S∞ − σ

λln S∞. (1.2)

Since I0 is usually very small, this equation is approximated by

S0 − σ

λln S0 = S∞ − σ

λln S∞. (1.3)

From the above summary for the model (1.1), we know that the disease dynamics of this modelis very clear: the disease either dies out quickly without causing new infectious, or experiencesa single outbreak before dying out. The final size S∞ and the magnitude of the outbreak (if any)depend on the initial susceptible population size S0. Obviously, this model does not include demo-graphic structure and is suitable for describing those diseases that suddenly develop in a communityand then disappear without infecting the entire community.

93


In this paper, we aim to modify the model (1.1) to include two important factors: latencyand spatial mobility. The scenario for the former is the fact that many diseases have latent periods,namely, an individual infected by a disease will not be infectious until some time after the infection.Among such diseases are Hepatitis B and tuberculosis including bovine tuberculosis, which maytake months to develop to infectious stage, as well as some childhood diseases such as measles,chicken pox, and whooping cough (pertussis), etc.(see Table 3.1 in [1]). For a disease with latency,it is reasonable to introduce into the model one more class: the exposed or latent class denotedby L(t), consisting of those individuals that have been infected but are not yet infectious. In sucha situation, the infection term λS(t)I(t) would not contribute directly to I ′(t); instead it shouldcontribute to L′(t). The inclusion of mobility in the model can be easily justified by the fact thatthe world has become highly connected nowadays, and travels between patches (towns, cities andcountries) have become more important and common, and sometimes even inevitable.

Along the above lines, there arise two basic questions: (Q1) how to incorporate these twofactors into model (1.1)? (Q2) what are the consequences? As an initial attempt, in this paperwe only consider a simple situation for these two factors: a fixed latent period and two patches.Under these two assumptions, we derive, in Section 2, a new model which is a modification of(1.1). The new model carries a discrete delay and non-local infection terms caused by the mobilityof latent individuals. Roughly speaking, such non-local terms explain the fact that an individualinfected in one of the two patches may be, due to his/her mobility, in either patch with certainprobability at the time when this individual becomes infectious. In Section 3, we are able toaddress the well-posedness of the model by proving the positivity and boundedness of solutions.By employing the theory of asymptotically autonomous systems, we are also able to show that, likein (1.1), the infectious populations in both patches will go to extinction, implying that the diseasewill eventually die out in both patches. Although we can not obtain two equations that determinethe two final sizes S1(∞) and S2(∞), we are able to prove that the ratio of these two final sizesare fully determined by the ratio of the dispersion rates of susceptible individuals between thetwo patches. The presence of the latent delay and the non-local infection terms in the modelmakes it very difficult to obtain a complete description of the patterns by which the disease diesout. However, our numeric study in Section 4 shows that the new model may demonstrate veryrich patterns by which the disease dies out. Particularly, multiple outbreaks may occur in the newmodel. This important finding is in contrast to the disease outbreak patterns described by the classicKermack-McKendrick model (1.1), and may provide some insights into the spread of diseases thatdo develop multiple outbreaks in reality. In Section 5, we summarize our results and discuss theirbiological implications; we also compare our model with some existing ones and discuss possiblefuture extension of the model to include a demographic structure.

We emphasize that the goal of this paper is to extend the classical Kermack-McKendrick model(1.1), which does not allow the persistence of the disease, to a model for a disease with a fixedlatent period spreading in a patchy environment. Thus, our model also does not support a diseasebecoming endemic. We point out that there have been numerous models that describe the diseasedynamics over patchy environments and that can support diseases either going to extinction orbecoming endemic, among which are [2, 3, 8, 12, 14, 15, 16] and the references therein. However,

94


these models do not consider a fixed latent period, neither do they contain non-local infection termswhich are exactly the novelty of our model (2.15).

2. Derivation of the modelConsider a population that lives in two patches. Let Si(t), Ii(t), Ri(t) be, respectively, the sub-populations of the susceptible, infectious and removed classes in patch i, i = 1, 2 at time t.The two patches are connected in the sense that individuals can move between these two patches.Assume that the disease has a fixed latent period, denoted by τ . Due to this latency and the mobilityof the individuals during the latent period, the rate at which patch 1 (patch 2) gains new infectiousat time t depends on the new infections infected τ time units ago not only in patch 1 (patch 2) butalso in patch 2 (patch 1). To determine such a dependence, we use the notion of infection age,denoted by a. Let li(t, a) be the density (with respect to the infection age a) of individuals at timet in patch i (i = 1, 2) with infection age a. Similar to the population with natural age structure(see Diekmann and Metz [9]), l1(t, a) and l2(t, a) are governed by the following first-order partialdifferential equations:

∂l1(t, a)

∂t+

∂l1(t, a)

∂a= −d1(a)l1(t, a) + D2(a)l2(t, a)−D1(a)l1(t, a),

∂l2(t, a)

∂t+

∂l2(t, a)

∂a= −d2(a)l2(t, a) + D1(a)l1(t, a)−D2(a)l2(t, a).

(2.1)

Here Dj(a)lj(t, a) corresponds to the dispersal of the individuals at the infection age a from patchj to patch i, where, 1 ≤ i 6= j ≤ 2, and di(a) represents the rate at which the infected individualsin patch i is removed via deaths due to the disease, recovery after treatments with permanentimmunity, and other possible means such as isolation or quarantine. In addition, we have assumedthat there is no delay in the dispersal between patches and there is no loss during migration frompatch j to patch i, meaning that all of those who leave patch j arrive at patch i safely. Obviously,if the two patches are too far away from each other, this assumption should be re-examined.

By the meaning of li(t, a), it is obvious that at a given time t, the number of the infectiousindividuals in patch i is given by

Ii(t) =

∫ ∞

τ

li(t, a)da. (2.2)

Obviously li(t, 0) corresponds to new infections which come from direct contacts between infec-tious and susceptible individuals. Mass action infection mechanism leads to

li(t, 0) = λiIi(t)Si(t). (2.3)

where λi is the the infection rate in patch i, i = 1, 2. It is also biologically reasonable to assume

li(t,∞) = 0. (2.4)

95


We point out that (2.4) holds if we assume that di(a), i = 1, 2, are bounded away from zero forlarge a (as is assumed in (2.5)). Indeed, using the method of characteristics, one can solve (2.1)and (2.3) with a given initial condition and directly verify that the solution satisfies (2.4).

For convenience of showing the main idea to build up the new model, we further assume that

di(a) =

{dl(a), for 0 ≤ a ≤ τ, and i = 1, 2,

dIi(a) = γi + µi, for a > τ, and i = 1, 2,

(2.5)

and

Di(a) =

{Dli(a) = Dli , for 0 ≤ a ≤ τ, and i = 1, 2,

DIi(a) = DIi

, for a > τ, and i = 1, 2,(2.6)

where dl(a) denotes the removal rate of latent individuals due to possible means such as quarantineor isolation, and it is assumed to be independent on the patch, and γi represents the removal rateof infectious individuals accounting for recovery with permanent immunity and possible isolationor quarantine in patch i, i = 1, 2, and µi corresponds to the disease mortality rate of infectiousindividuals in patch i, i = 1, 2. For simplicity of notation, in the sequel, we let σi = γi + µi,i = 1, 2.

By integrating (2.1) with respect to a from τ to ∞, we have

dIi(t)

dt= −

∫ ∞

τ

∂li(t, a)

∂ada−

∫ ∞

τ

dIi(a)li(t, a)da

+

∫ ∞

τ

DIj(a)lj(t, a)da−

∫ ∞

τ

DIi(a)li(t, a)da

= li(t, τ)− σiIi(t) + DIjIj(t)−DIi

Ii(t), (2.7)

for i, j = 1, 2 with j 6= i. The equations governing Si(t) and Ri(t) are given in the usual way,which, together with (2.7), lead to the following model under the above assumptions:

dS1(t)

dt= DS2S2(t)−DS1S1(t)− λ1I1(t)S1(t),

dS2(t)


dI1(t)

dt= −σ1I1(t) + DI2I2(t)−DI1I1(t) + l1(t, τ),

dI2(t)

dt= −σ2I2(t) + DI1I1(t)−DI2I2(t) + l2(t, τ),

dR1(t)

dt=

∫ τ

0

dl(a)l1(t, a)da + σ1I1(t) + DR2R2(t)−DR1R1(t),

dR2(t)

dt=

∫ τ

0

dl(a)l2(t, a)da + σ2I2(t) + DR1R1(t)−DR2R2(t).

(2.8)

where DSi≥ 0 is the rate at which susceptible individuals migrate from patch i to patch j (j 6= i),

and DRi≥ 0 is the rate at which removed individuals migrate from patch i to patch j (j 6= i).

96


Note that as in the classic Kermack-McKendrick model (1.1), the two equations for the re-moved classes R1 and R2 are decoupled from the equations for S1, S2, I1 and I2, thus we onlyneed to consider the first four equations in (2.8). The remaining work in this section is to evaluatethe term li(t, τ) for i = 1, 2 in terms of Si and Ii (i = 1, 2). To this end, we fix s ≥ 0 and let

V si (t) = li(t, t− s), for s ≤ t ≤ s + τ and i = 1, 2.

Then, for 1 ≤ i 6= j ≤ 2,

d

dtV s

i (t) =∂

∂tli(t, a)|a=t−s +

∂

∂ali(t, a)|a=t−s

= −di(t− s)li(t, t− s) + Dj(t− s)lj(t, t− s)−Di(t− s)li(t, t− s)

= −dl(t− s)V si (t) + Dlj(t− s)V s

j (t)−Dli(t− s)V si (t). (2.9)

Since s ≤ t ≤ s + τ , we have

d

dt

(V s

1 (t) + V s2 (t)

)= −dl(t− s)

(V s

1 (t) + V s2 (t)

). (2.10)

Solving this linear equation and using the condition (2.3), we get

V s1 (t) + V s

2 (t) = e−R t

s dl(θ−s)dθ(V s

1 (s) + V s2 (s)

)

= e−R t

s dl(θ−s)dθ(l1(s, 0) + l2(s, 0)

)

= e−R t

s dl(θ−s)dθ(λ1I1(s)S1(s) + λ2I2(s)S2(s)

)

= e−R t−s0 dl(a)da

(λ1I1(s)S1(s) + λ2I2(s)S2(s)

), for s ≤ t ≤ s + τ. (2.11)

Hence, from (2.9),

d

dtV s

1 (t) = −dl(t− s)V s1 (t) + Dl2(t− s)V s

2 (t)−Dl1(t− s)V s1 (t)

= −dl(t− s)V s1 (t) + Dl2(t− s)

(V s

1 (t) + V s2 (t)

)− (Dl1(t− s) + Dl2(t− s)

)V s

1 (t)

= −(dl(t− s) + Dl1(t− s) + Dl2(t− s)

)V s

1 (t)

+Dl2(t− s)e−R t−s0 dl(a)da

(λ1I1(s)S1(s) + λ2I2(s)S2(s)

)

= −D∗(t− s)V s1 (t) + Dl2(t− s)e−

R t−s0 dl(a)da

(λ1I1(s)S1(s) + λ2I2(s)S2(s)

),

where D∗(a) , dl(a) + Dl1(a) + Dl2(a). This is a first-order linear inhomogeneous equation for

97


V s1 (t), the solution of which is given, using (2.3), by

V s1 (t) = e−

R ts D∗(θ−s)dθV s

1 (s)

+∫ t

se−

R tξ D∗(θ−s)dθDl2(ξ − s)e−

R ξ−s0 dl(a)dadξ

(λ1I1(s)S1(s) + λ2I2(s)S2(s)

)

= e−R t−s0 D∗(a)daλ1I1(s)S1(s)

+∫ t

se−

R t−sξ−s D∗(a)daDl2(ξ − s)e−

R ξ−s0 dl(a)dadξ

(λ1I1(s)S1(s) + λ2I2(s)S2(s)

).

Let D̂(a) = Dl1(a) + Dl2(a). Then

l1(t, τ) = V t−τ1 (t)

=(e−

R τ0 D∗(a)da +

∫ t

t−τ

e−R τ

ξ−t+τ D∗(a)daDl2(ξ − t + τ)e−R ξ−t+τ0 dl(a)dadξ

)λ1I1(t− τ)S1(t− τ)

+( ∫ t

t−τ

e−R τ

ξ−t+τ D∗(a)daDl2(ξ − t + τ)e−R ξ−t+τ0 dl(a)dadξ

)λ2I2(t− τ)S2(t− τ)

=(e−

R τ0 D∗(a)da + e−

R τ0 dl(a)da

∫ τ

0

e−R τ

θ D̂(a)daDl2(θ)dθ)λ1I1(t− τ)S1(t− τ)

+(e−

R τ0 dl(a)da

∫ τ

0

e−R τ


= e−R τ0 dl(a)da

(1−

∫ τ

0

e−R τ


+(e−

R τ0 dl(a)da

∫ τ

0

e−R τ

θ D̂(a)daDl2(θ)dθ)λ2I2(t− τ)S2(t− τ).

Thus,

l1(t, τ) = ε(1− α1(τ)

)λ1I1(t− τ)S1(t− τ) + εα2(τ)λ2I2(t− τ)S2(t− τ), (2.12)

where

ε = e−R τ0 dl(a)da, α1(τ) =

∫ τ

0

e−R τ

θ D̂(a)daDl1(θ)dθ, α2(τ) =

∫ τ

0

e−R τ

θ D̂(a)daDl2(θ)dθ.

(2.13)Similarly,

l2(t, τ) = ε(1− α2(τ)

)λ2I2(t− τ)S2(t− τ) + εα1(τ)λ1I1(t− τ)S1(t− τ). (2.14)

98


Substituting (2.12) and (2.14) into (2.8) and taking out the first four equations for S1, S2, I1

and I2 results in the following new model:

dS1(t)


dS2(t)


dI1(t)

dt= −σ1I1(t) + DI2I2(t)−DI1I1(t)

+ε(1− α1(τ)

)λ1I1(t− τ)S1(t− τ) + εα2(τ)λ2I2(t− τ)S2(t− τ),

dI2(t)

dt= −σ2I2(t) + DI1I1(t)−DI2I2(t)

+ε(1− α2(τ)

)λ2I2(t− τ)S2(t− τ) + εα1(τ)λ1I1(t− τ)S1(t− τ).

(2.15)

From the above model, we see that the dispersion of latent individuals plays a different rolefrom that of susceptible and infectious individuals. The explanation for those instantaneous termsare quite straightforward, and we now explain those delayed terms in the model. The probabilitythat an individual infected in patch 1 can survive to the infection age τ (at which the infectedindividual becomes infectious) is ε. Due to the mobility during the latent period between the twopatches, τ time units later, a survived infected individual infected in patch 1, may be in patch 1with probability

(1 − α1(τ)

)or in patch 2 with probability α1(τ). This explains the term ε

(1 −

α1(τ))λ1I1(t− τ)S1(t− τ) in I1 equation and the term εα1(τ)λ1I1(t− τ)S1(t− τ) in I2 equation.

The terms ε(1−α2(τ)

)λ2I2(t−τ)S2(t−τ) in I2 equation and the term εα2(τ)λ2I2(t−τ)S2(t−τ) in

I1 equation are explained similarly. Alternatively, we may explain these terms in light of fractionsas below. Among the individuals infected in the first patch τ time units ago, a fraction ε can surviveto the infection age τ , a fraction

(1 − α1(τ)

)of which is now in patch 1 while a fraction α1(τ) is

in patch 2.

3. Mathematical analysisModel (2.15) is a system of delay differential equations with non-local interactions over a two-patch environment. Firstly and naturally, as far as mathematical analysis of the system is con-cerned, initial conditions of delayed type should be considered. In other words, we consider thesystem with the following initial conditions:

Si(θ) = φi(θ) ≥ 0, and Ii(θ) = ψi(θ) ≥ 0, for i = 1, 2, and θ ∈ [−τ, 0], (3.1)

where nonnegativity is based on biological consideration. By the fundamental theory of delay dif-ferential equations (see, e.g., Hale and Verduyn Lunel [6]), we know that for any given continuousfunctions φi(θ), ψi(θ), i = 1, 2, the initial value problem (2.15) with (3.1) has a unique solution.In order for (2.15) with (3.1) to be biologically well-posed, we need to make sure that the solutionremains nonnegative for t ≥ 0 and is bounded.

99


Theorem 1. Assume that φi(θ) and ψi(θ), i = 1, 2, are continuous. Then the solution of the initialvalue problem (2.15) with (3.1) remains nonnegative for t ≥ 0 and is bounded.

Proof. Firstly, we show the nonnegativity of the solution. For this purpose, let us rewrite thesystem (2.15) as follows:

d

dtS(t) = M(t)S(t), (3.2)

d

dtI(t) = AI(t) + B(t)I(t− τ), (3.3)

where S(t) = (S1(t), S2(t))T and I(t) = (I1(t), I2(t))

T with T representing the transpose of avector, and

M(t) ,[ −DS1 − λ1I1(t) DS2

DS1 −DS2 − λ2I2(t)

],

A ,[ −(σ1 + DI1) DI2

DI1 −(σ2 + DI2)

],

B(t) ,[

ε(1− α1(τ)

)λ1S1(t− τ) εα2(τ)λ2S2(t− τ)

εα1(τ)λ1S1(t− τ) ε(1− α2(τ)

)λ2S2(t− τ)

].

Noting that the off-diagonal elements of matrix M(t) are nonnegative, we conclude that the entriesof the matrix e

R t0 M(ξ)dξ are all nonnegative. Indeed, let h(t) = max{DS1 + λ1I1(t) + 1, DS2 +

λ2I2(t) + 1} and rewrite M(t) as

M(t) =

[ −h(t) 0

0 −h(t)

]+

[h(t)−DS1 − λ1I1(t) DS2

DS1 h(t)−DS2 − λ2I2(t)

], −h(t)E+M1(t),

where E is the 2× 2 identity matrix. Then all entries of M1(t) are nonnegative, and hence, so arethe entries of e

R t0 M1(ξ)dξ. Also

eR t0 −h(ξ)Edξ =

e−

R t0 h(ξ)dξ 0

0 e−R t0 h(ξ)dξ

.

Noting that the scalar matrix −h(t)E commutes with any 2 × 2 matrix (hence with M1(t)), wehave

eR t0 M(ξ)dξ = e

R t0 −h(ξ)Edξe

R t0 M1(ξ)dξ,

implying that all entries of eR t0 M(ξ)dξ are nonnegative. Now, from the equation (3.2), we obtain

S(t) = eR t0 M(ξ)dξS(0) ≥ 0, for t ≥ 0, (3.4)

100


Similarly, for any t ≥ 0, all entries of eAt are nonnegative. Moreover, by the nonnegativity ofS1(t) and S2(t) established above, we know that all entries of B(t) are all nonnegative. Now, (3.3)leads to

I(t) = eAtI(0) +

∫ t

0

eA(t−ξ)B(ξ)I(ξ − τ)dξ, (3.5)

implying I(t) ≥ 0 for t ∈ [0, τ ] from the initial condition Ii(θ) ≥ 0 for θ ∈ [−τ, 0] and i = 1, 2.This and (3.5) ensure I(t) ≥ 0 for t ∈ [τ, 2τ ]. By induction, we then conclude that I(t) ≥ 0 fort ≥ 0.

Now, we show that Si(t) and Ii(t) for i = 1, 2 are bounded. Let N(t) = S1(t)+I1(t)+S2(t)+I2(t). Direct calculation shows that

d

dtN(t) = −σ1I1(t)− σ2I2(t)− λ1I1(t)S1(t)− λ2I2(t)S2(t)

+ελ1I1(t− τ)S1(t− τ) + ελ2I2(t− τ)S2(t− τ)

≤ −σ1I1(t)− σ2I2(t)− λ1I1(t)S1(t)− λ2I2(t)S2(t)

+λ1I1(t− τ)S1(t− τ) + λ2I2(t− τ)S2(t− τ)

= −σ1I1(t)− σ2I2(t)− d

dt

∫ t

t−τ

λ1I1(ξ)S1(ξ)dξ − d

dt

∫ t

t−τ

λ2I2(ξ)S2(ξ)dξ

= −σ1I1(t)− σ2I2(t)− d

dt

∫ t

t−τ

(λ1I1(ξ)S1(ξ) + λ2I2(ξ)S2(ξ)

)dξ.

Hence,

d

dt

(N(t) +

∫ t

t−τ

(λ1I1(ξ)S1(ξ) + λ2I2(ξ)S2(ξ)

)dξ

)≤ −σ1I(t)− σ2I2(t) ≤ 0,

implying

N(t) +

∫ t

t−τ

(λ1I1(ξ)S1(ξ) + λ2I2(ξ)S2(ξ)

)dξ ≤ N(0) +

∫ 0

−τ

(λ1I1(ξ)S1(ξ) + λ2I2(ξ)S2(ξ)

)dξ.

By nonnegativity of Si(t) and Ii(t) (i = 1, 2), we have

N(t) ≤ N(0) +

∫ 0

−τ

(λ1I1(ξ)S1(ξ) + λ2I2(ξ)S2(ξ)

)dξ

= S1(0) + S2(0) + I1(0) + I2(0) +

∫ 0

−τ

(λ1I1(ξ)S1(ξ) + λ2I2(ξ)S2(ξ)

)dξ , K. (3.6)

Therefore,

Si(t), Ii(t) ≤ K, i = 1, 2, (3.7)

101


where K is the constant on the right-hand side of (3.6), depending on the initial values of suscep-tible and infectious individuals. The proof of Theorem 1 is completed.

Remark 2. From (3.4) and (3.5), we know that (i) if either S1(0) > 0 or S2(0) > 0, then bothS1(t) and S2(t) remain strictly positive for all t > 0; (ii) if either I1(0) > 0 or I2(0) > 0, thenboth I1(t) and I2(t) remain strictly positive for all t > 0.

Next, we show that, as in the classic Kermack-McKendrick model, the I1(t) and I2(t) compo-nents of the solution to (2.15) with (3.1) always converge to zero.

Let G(t) = S1(t) + S2(t). Then

d

dtG(t) = −(

λ1I1(t)S1(t) + λ2I2(t)S2(t)) ≤ 0, (3.8)

and hence, G(t) is a decreasing function. By this monotonicity and Theorem 1, we conclude thatG(∞) = limt→∞ G(t) exists and is nonnegative, and limt→∞ G′(t) = 0. This and (3.8) in turnimply that

λ1I1(t)S1(t) + λ2I2(t)S2(t) → 0, as t →∞. (3.9)

This limit, together with the nonnegativity of Si(t) and Ii(t) (i = 1, 2) leads to

I1(t)S1(t) → 0, I2(t)S2(t) → 0, as t →∞. (3.10)

Now rewrite (3.3) asd

dtI(t) = AI(t) + P (t), (3.11)

where P (t) = (p1(t), p2(t))T with

p1(t) = ε(1− α1(τ)

)λ1S1(t− τ)I1(t− τ) + εα2(τ)λ2S2(t− τ)I2(t− τ) → 0,

p2(t) = εα1(τ)λ1S1(t− τ)I1(t− τ) + ε(1− α2(τ)

)λ2S2(t− τ)I2(t− τ) → 0,

as t →∞ by (3.10). Thus, (3.11) has

d

dtX(t) = AX(t) (3.12)

as its limit equation for which the trivial equilibrium (0, 0) is globally asymptotically stable. Bythe theory of asymptotically autonomous systems (see, e.g., [10, 13]), every solution I(t) of (3.11)also converges to (0, 0) as t →∞. Thus, we have proved

Theorem 3. Let (S1(t), S2(t), I1(t), I2(t)) be the solution of (2.15) with (3.1). Then, I1(t) → 0and I2(t) → 0 as t →∞.

102


Next, we address the long term behavior of S1(t) and S2(t). In the classic Kermack-McKendrickmodel (1.1) where the susceptible population is immediately seen to be decreasing (and bounded)implying that there is a final size S(∞) = limt→∞ S(t). However, for (2.15), the existence oflimt→∞ S1(t) and limt→∞ S2(t) is not obvious and needs to be confirmed.

Let F (t) = DS2S2(t)−DS1S1(t). From (2.15), we have

dF

dt= −(DS1 + DS2)F (t) + r(t), (3.13)

where r(t) = DS1λ1I1(t)S1(t) − DS2λ2I2(t)S2(t). By (3.10), r(t) → 0 as t → ∞, and hence,equation (3.13) has

df(t)

dt= −(DS1 + DS2)f(t) (3.14)

as its limit equation whose dynamics is global convergence to 0. By the theory of asymptoticallyautonomous systems (see, e.g., [10, 13]), F (t) converges to 0 as t → ∞. Note that G(t) =S1(t) + S2(t) also converges as t →∞ (see (3.8)). Now by

S1(t) =DS2G(t)− F (t)

DS1 + DS2

and S2(t) =F (t) + DS1G(t)

DS1 + DS2

, (3.15)

we conclude that both S1(∞) = limt→∞ S1(t) and S2(∞) = limt→∞ S2(t) exist. Moreover, theequation (3.15) and the limit F (∞) = 0 lead to

S1(∞) =DS2

DS1 + DS2

G(∞) and S2(∞) =DS1

DS1 + DS2

G(∞), (3.16)

which determine the ratio of the two final sizes:

S1(∞)

S2(∞)=

DS2

DS1

. (3.17)

Summarizing the above, we have established the following

Theorem 4. Let (S1(t), S2(t), I1(t), I2(t)) be the solution of (2.15) with (3.1). Then, S1(∞) =limt→∞ S1(t) and S2(∞) = limt→∞ S2(t) exist and satisfy (3.17).

Although Theorems 3-4 have confirmed that the solution (S1(t), S2(t), I1(t), I2(t)) of (2.15)with (3.1) converges to (S1(∞), S2(∞), 0, 0) with the ratio S1(∞)/S2(∞) being determined by thedispersion coefficients of the susceptible sub-populations between the two patches, unfortunately,we are unable to derive an equation parallel to (1.2) that determines the final sizes S1(∞) andS2(∞), even implicitly. This is because, unlike (1.1) for which one can easily find a first integral,for the generalized Kermack-McKendrick model (2.15), integration becomes very difficult, if notimpossible.

103


Another regret for (2.15) is that we are unable to determine the patterns by which the diseasedies out in both patches. This is because there are more terms, including both instantaneous anddelayed terms, on the right-hand sides of the I1(t) and I2(t) equations. In the next section, we willperform some numeric simulations which show that the new model (2.15) may demonstrate veryrich patterns for the disease to die out. In particular, it allows multiple outbreaks, contrasting to thesimple outbreak pattern described by the classic Kermack-McKendrick model (1.1).

4. Numeric study of disease patternsIn this section, we numerically explore the patterns by which the disease dies out. The mainfocus is the impact of the latent period τ and/or dispersion rates on the outbreak patterns. For thispurpose, we will discuss the following two cases: the case in which the disease does not have alatency, and the case in which the disease does have a latency. In the first case, we will observehow the dispersion of susceptible and infectious individuals affects the patterns of disease outbreakin both patches. In the latter case, we will investigate how the length of the latent period influencesthe disease dynamics when the dispersion rates are fixed, and how the dispersal rates of the latentindividuals affect the dynamics of the disease with a fixed latent period. Finally, we will investigatenumerically the joint effect of the disease latency and spatial dispersion on the disease outbreakpatterns.

Since our focus is the impact of dispersion and latency, throughout this section, we set theparameters for the removal rates and disease transmission rates in both patches as follows:

σ1 = 0.55, σ2 = 0.75, λ1 = 0.0005, λ2 = 0.00025. (4.1)

Without loss of generality, we only consider the simpler case with dl = 0 giving ε = 1. Forconvenience of simulations, we calculate the non-local factors α1(τ) and α2(τ) defined in (2.13)as the following more explicit formulas:

α1(τ) =Dl1

Dl1 + Dl2

(1− e−(Dl1

+Dl2)τ

), α2(τ) =

Dl2

Dl1 + Dl2

(1− e−(Dl1

+Dl2)τ

). (4.2)

4.1. The case without latency: τ = 0

When the disease has no latency, τ = 0 and hence α1 = α2 = 0. In such a simple case, the modelreduces to

d

dtS1(t) = DS2S2(t)−DS1S1(t)− λ1I1(t)S1(t),

d

dtS2(t) = DS1S1(t)−DS2S2(t)− λ2I2(t)S2(t),

d

dtI1(t) = −σ1I1(t) + DI2I2(t)−DI1I1(t) + λ1I1(t)S1(t),

d

dtI2(t) = −σ2I2(t) + DI1I1(t)−DI2I2(t) + λ2I2(t)S2(t).

(4.3)

104


We are interested in how the dispersion of susceptible and infectious individuals influences the pat-terns by which the disease dies out in both patches. For this purpose, we distinguish the followingtwo cases: one is that susceptible individuals travel but infectious individuals in both patches donot travel, the other is that individuals of both classes in both patches travel.

4.1.1. Susceptible individuals travel but infectious individuals of both patches do not travel

This case corresponds to DIi= 0, (i = 1, 2) and DSi

> 0, (i = 1, 2). In such a situation, fromthe third and fourth equations in (4.3), it is seen that, just as in the classical Kermack-McKendrickmodel, there is a critical value S∗0 = σ/λ in each patch for the initial size of the susceptible class,beyond which the disease in that patch will initially experience an outbreak (I(t) increases), andbelow which the disease initially decays. With (4.1), these two critical values are S∗10 = σ1/λ1 =1100 and S∗20 = σ2/λ2 = 3000. Unlike in the classical Kermack-McKendrick model where S(t)always decreases, now with the positive dispersion rates DSi

> 0, (i = 1, 2), S1(t) and/or S2(t)may experience some growth initially, although S1(t) + S2(t) always decreases. This will causechanges in the disease dynamics in the two patches.

Figure 1 shows the phase portraits of Ii vs Si, i = 1, 2 for various sets of DS1 and DS2 valuesshown in its caption. The initial values are given as follows:

S10 = 3000, S20 = 2500, I10 = 10, I20 = 50. (4.4)

The ratio d := DS1/DS2 describes the relative strengths of the dispersion of susceptible individualsbetween the two patches, and the four subfigures are results of numeric simulations obtained byvarying the value of d ascending order. Note that S10 > S∗10 and S20 < S∗20 . Thus, I1(t) initiallyincreases but I2(t) initially decreases. It is interesting to notice that when d is reasonably large,I2(t) will also experience an outbreak before it dies out, as is shown in Figure 1-(iii)-(iv). This isbecause when DS1 is much larger than DS2, and hence, there will be more susceptible individualstraveling from Patch 1 to Patch 2 than from Patch 2 to Patch 1, making more available susceptibleindividuals in Patch 2 for infection. Therefore, such an outbreak in Patch 2 is purely caused by theunbalanced travel of susceptible individuals.

Now we change the initial distributions (4.4) in both patches to

S10 = 3000, S20 = 3500, I10 = 10, I20 = 50, (4.5)

such that both I1(t) and I2(t) increase initially. This time, varying the ratio d := DS1/DS2 doesnot change the patterns of I1 and I2 (one outbreak), but only causes some changes to the patternsof S1 and S2. However, simulations show that the change of d does affect the magnitudes of theoutbreaks in the two patches, as shown in Figure 2.

4.1.2. Susceptible and infectious individuals in both patches travel

In this case, common sense seems to suggest the assumption DSi> DIi

, (i = 1, 2), since infectiousindividuals have a lower level of travel intention. We choose DI2 = 0.10 > DI1 = 0.09. From

105


the third and fourth equations in (4.3), larger DI2 should boost I1 initially but should not favor I2.Numeric simulations show that, again as in Section 4.1.1, larger values of DS1 (relative to DS2)can cause an outbreak of the disease in Patch 2. See Figure 3 for simulations of (4.3) with initialvalues (4.4).

Figure 4 are some numeric simulation results for (4.3) but with initial distribution (4.5). Com-paring the Figure 4-(ii) with Figure 2-(ii), we see that in the former, I2(t) first experiences a quies-cent period before its outbreak. This is due to the interplay of DIi

and DSi, i = 1.2.

4.2. The cases with latency τ > 0

In this case, the model is a system of delay differential equations. We mainly examine the impactof the latent delay τ and the dispersion rates Dli , i = 1, 2, during the latent period. To this end,we first fix τ > 0, DSi

, and DIi, i = 1, 2, at some levels, and observe how the disease patterns

will change as Dli , i = 1, 2, vary. Then, we will fix the dispersion rates at certain levels andnumerically investigate the impact of the latent delay τ on the disease dynamics. When we setvalues of the dispersion rates for simulations, for realistic reasons, we abide by the restrictionDIi

≤ Dli ≤ DSi, i = 1, 2.

For simulation convenience, we introduce the ratio l := Dl1/Dl2 , which measures the relativedispersion strength of latent individuals in Patch 1 to Patch 2.

4.2.1. For fixed τ > 0 consider the impact of various dispersion rates

In this subsection, we fixed the disease latency at τ = 10. Figure 5 corresponds to the case inwhich the infectious individuals do not travel and Figure 6 are simulation results of the case whenthe infectious individuals travel but at a lower rate than the latent individuals do. In both figures,the initial functions are taken constant functions with values given by (4.5) on [−τ, 0].

Figure 5-(i) shows the disease dynamics in the absence of dispersion (two patches are discon-nected) where the disease experiences one outbreak in both patches before dying out, although theoutbreak speed in Patch 1 fluctuates at several points. The incorporation of dispersion for latentand susceptible classes has an immediate impact on the pattern in Patch 2: I2(t) changes frombeing initially increasing to being initial decreasing before going to an outbreak, as are shown inFigure 5-(ii)-(iii)-(iv). Such a change in the initial disease dynamics may easily give one a wrongimpression that the disease starts dying out, and hence is worth particular attention and warning.Another important finding is that the dispersion strengths of latent individuals may cause multipleoutbreaks, as is shown in Figure 5-(iv) where the disease evolves into two outbreaks in both patchesfrom the previous one outbreak.

In Figure 6-(ii)-(iii)-(iv), different from Figure 5, DI1 and DI2 are fixed at positive values DI1 =0.1 and DI2 = 0.09. Similar phenomena to that in Figure 5 are observed, but the two-outbreakpattern develops earlier than that in Figure 5 as l is increased (it appears for l = Dl1/Dl2 = 1 inFigure 6 but not in Figure 5). This implies that DIi

, i = 1, 2, also play a role in causing multipleoutbreaks.

106


4.2.2. Impact of the length of the latent period

In this subsection, we numerically investigate the influence of the magnitude of the disease latencyon the pattern of the disease outbreak. For this purpose, we fix the dispersion rates at several levelsas in the previous subsections, indicated as subfigures (i)-(iv) as before, and observe the changes ofthe disease patterns for three values of τ : τ = 5, 10, 15. The initial functions are taken constantfunctions with values given by (4.5) on [−τ, 0].

Figure 7-(i) and Figure 8-(i) reveals that even if the two patches are isolated, larger latencytends to cause multiple outbreaks and in the mean time, decrease the outbreak sizes. Figure 7-(ii)-(iii)-(iv) show that the two patches may have different number of outbreaks, due to the interactionof DS1 and DS2 . For example, in Figure 7-(ii) and for τ = 15, Patch 2 only experiences oneoutbreak while Patch 1 suffers two outbreaks.

Figure 8 aims at comparing the results of various values of the ratio l = Dl1/Dl2 . Interestingly,when Dli and DIi

are turned on, the inconsistency in the numbers of outbreaks in the two patchesdisappears, as is shown in Figure 8-(ii)-(iii)-(iv) (comparing with 7-(ii)-(iii)-(iv)). Also in Figure8-(ii)-(iii)-(iv), for the given values of the parameters and initial values, a relatively longer periodof quiescence for the disease in Patch 2 is observed, after an initial decrease and before its firstoutbreak. This quiescent period seems to increase as τ and l = Dl1/Dl2 increase.

Another observation from Figure 7 and 8 is that although a larger latent period can causemultiple outbreaks, it also decreases the sizes of outbreaks. The dispersion rates also affect theoutbreak sizes. Analytically determining these sizes seem to be very difficult, if not impossible.

At last, we can also numerically confirm the conclusion on the final sizes in the two patchesgiven in Theorem 4. For example, consider the model with parameters and initial values leading toFigure 7. At t = 200, Ii(200) ≈ 0. For d = 0.1, we have S1(200) = 102.6957, S2(200) = 10.2696,and hence S2(200)/S1(200) = 0.1000 = d; for d = 1, we have S1(200) = 385.8986, S2(200) =385.8986, and S2(200)/S1(200) = 1; for d = 2, we have S1(200) = 576.4, S2(200) = 1152.7,and S2(200)/S1(200) = 2.0000, all confirming the relation S2(∞)/S1(∞) = DS1/DS2 = d. Forthe parameters and initial values leading to Figure 8, d = DS1/DS2 = 0.45/0.5 = 0.9000 is fixedin Figure 8-(ii)-(iii)-(iv). Now for l = 0.1, S1(200) = 295.3050 and S2(200) = 265.7736 givingS2(200)/S1(200) = 0.9000 = d; for l = 0.5, S1(200) = 382.4305 and S2(200) = 344.1852 alsogiving S2(200)/S1(200) = 0.9000 = d; for l = 1.5, S1(200) = 662.5170 and S2(200) = 596.2653again giving S2(200)/S1(200) = 0.9000 = d.

107


0 1000 2000 30000

200

400

600

800

1000

S1(t)

I 1(t

)

Patch 1 when isolated

2000 2500 30000

10

20

30

40

50

60

S2(t)

I 2(t

)


0 2000 40000

500

1000

1500

2000

S1(t)

I 1(t

)

Patch 1 with d=0.1

0 1000 2000 30000

10

20

30

40

50

60

S2(t)

I 2(t

)

Patch 2 with d=0.1

0 1000 2000 30000

50

100

150

200

250

300

S1(t)

I 1(t

)

Patch 1 with d=2

1000 2000 3000 40000

20

40

60

80

100

S2(t)

I 2(t

)

Patch 2 with d=2

0 1000 2000 30000

5

10

15

20

25

30

S1(t)

I 1(t

)

Patch 1 with d=4

0 2000 4000 60000

50

100

150

200

250

300

350

400

S2(t)

I 2(t

)

Patch 2 with d=4

(i) (ii) (iii) (iv)

Figure 1: The phase portraits of Ii(t) vs Si(t)(i = 1, 2) for (4.3) with (4.1), DIi = 0, i = 1, 2 and (i)DS1 = 0, DS2 = 0; (ii) DS1 = 0.02, DS2 = 0.2; (iii) DS1 = 0.4, DS2 = 0.2; (iv) DS1 = 0.8, DS2 =0.2. Initial values are given by (4.4).

108


0 1000 2000 30000

200

400

600

800

1000

S1(t)

I 1(t

)


0 2000 40000

500

1000

1500

2000

2500

3000

S1(t)

I 1(t

)

Patch 1 with d=0.1

0 2000 40000

500

1000

1500

S1(t)

I 1(t

)

Patch 1 with d=1

0 1000 2000 30000

100

200

300

400

500

600

S1(t)

I 1(t

)

Patch 1 with d=2

2000 2500 3000 35000

20

40

60

80

100

S2(t)

I 2(t

)


0 2000 40000

10

20

30

40

50

60

S2(t)

I 2(t

)

Patch 2 with d=0.1

0 2000 40000

20

40

60

80

100

S2(t)

I 2(t

)

Patch 2 with d=1

1000 2000 3000 40000

50

100

150

200

250

300

S2(t)

I 2(t

)

Patch 2 with d=2

(i) (ii) (iii) (iv)

Figure 2: The phase portraits of Ii(t) vs Si(t)(i = 1, 2) for (4.3) with (4.1), DIi = 0, i = 1, 2 and (i)DS1 = 0, DS2 = 0; (ii) DS1 = 0.02, DS2 = 0.2; (iii) DS1 = 0.2, DS2 = 0.2; (iv) DS1 = 0.4, DS2 =0.2. Initial values are given by (4.5).

109


0 1000 2000 30000

200

400

600

800

1000

S1(t)

I 1(t

)


0 2000 40000

500

1000

1500

S1(t)

I 1(t

)

Patch 1 with d=0.1

0 2000 40000

200

400

600

800

1000

S1(t)

I 1(t

)

Patch 1 with d=0.8

0 1000 2000 30000

50

100

150

200

250

300

S1(t)

I 1(t

)

Patch 1 with d=2

2000 2500 30000

10

20

30

40

50

60

S2(t)

I 2(t

)


0 1000 2000 30000

50

100

150

200

S2(t)

I 2(t

)

Patch 2 with d=0.1

0 1000 2000 30000

50

100

150

S2(t)

I 2(t

)

Patch 2 with d=0.8

1000 2000 3000 40000

50

100

150

S2(t)

I 2(t

)

Patch 2 with d=2

(i) (ii) (iii) (iv)

Figure 3: The phase portraits of Ii(t) vs Si(t)(i = 1, 2) for (4.3) with (4.1) and (i) DS1 = 0, DS2 =0, DI1 = 0, DI2 = 0; (ii) DS1 = 0.02, DS2 = 0.2, DI1 = 0.09, DI2 = 0.1; (iii) DS1 = 0.16, DS2 =0.2, DI1 = 0.09, DI2 = 0.1; (iv) DS1 = 0.4, DS2 = 0.2, DI1 = 0.09, DI2 = 0.1. Initial values aregiven by (4.4).

110


0 1000 2000 30000

200

400

600

800

1000

S1(t)

I 1(t

)


0 2000 40000

500

1000

1500

2000

S1(t)

I 1(t

)

Patch 1 with d=0.1

0 2000 40000

200

400

600

800

1000

S1(t)

I 1(t

)

Patch 1 with d=1

0 1000 2000 30000

100

200

300

400

500

600

S1(t)

I 1(t

)

Patch 1 with d=2

2000 2500 3000 35000

20

40

60

80

100

S2(t)

I 2(t

)


0 2000 40000

50

100

150

200

250

300

S2(t)

I 2(t

)

Patch 2 with d=0.1

0 2000 40000

50

100

150

200

250

300

S2(t)

I 2(t

)

Patch 2 with d=1

1000 2000 3000 40000

50

100

150

200

250

300

S2(t)

I 2(t

)

Patch 2 with d=2

(i) (ii) (iii) (iv)

Figure 4: The phase portraits of Ii(t) vs Si(t)(i = 1, 2) for (4.3) with (4.1) and (i) DS1 = 0, DS2 =0, DI1 = 0, DI2 = 0; (ii) DS1 = 0.02, DS2 = 0.2, DI1 = 0.09, DI2 = 0.1; (iii) DS1 = 0.2, DS2 =0.2, DI1 = 0.09, DI2 = 0.1; (iv) DS1 = 0.4, DS2 = 0.2, DI1 = 0.09, DI2 = 0.1. Initial values aregiven by (4.5).

111


0 2000 40000

50

100

150

200

250

300

S1(t)

I 1(t

)


0 2000 40000

50

100

150

200

250

300

350

400

S1(t)

I 1(t

)

Patch 1 with l=0.1

0 2000 40000

50

100

150

200

250

S1(t)

I 1(t

)

Patch 1 with l=0.5

0 2000 40000

20

40

60

80

100

120

S1(t)

I 1(t

)

Patch 1 with l=1.5

0 2000 40000

10

20

30

40

50

60

S2(t)

I 2(t

)


0 2000 40000

10

20

30

40

50

S2(t)

I 2(t

)

Patch 2 with l=0.1

0 2000 40000

10

20

30

40

50

60

70

80

S2(t)

I 2(t

)

Patch 2 with l=0.5

0 2000 40000

20

40

60

80

100

120

S2(t)

I 2(t

)

Patch 2 with l=1.5

(i) (ii) (iii) (iv)

Figure 5: The phase portraits of Ii(t) vs Si(t) (i = 1, 2) for (2.15) with (4.1), ε = 1, DIi = 0, i = 1, 2and (i) DS1 = 0, DS2 = 0, Dl1 = 0, Dl2 = 0; (ii) DS1 = 0.45, DS2 = 0.5, Dl1 = 0.03, Dl2 = 0.3; (iii)DS1 = 0.45, DS2 = 0.5, Dl1 = 0.15, Dl2 = 0.3; (iv) DS1 = 0.45, DS2 = 0.5, Dl1 = 0.45, Dl2 = 0.3.Initial values are given by (4.5).

112


0 1000 2000 30000

50

100

150

200

S1(t)

I 1(t

)


0 2000 40000

50

100

150

200

250

300

350

S1(t)

I 1(t

)

Patch 1 with l=0.1

0 2000 40000

50

100

150

S1(t)

I 1(t

)

Patch 1 with l=1

0 2000 40000

20

40

60

80

100

120

S1(t)

I 1(t

)

Patch 1 with l=1.5

1000 2000 3000 40000

10

20

30

40

50

60

70

S2(t)

I 2(t

)


0 2000 40000

10

20

30

40

50

60

S2(t)

I 2(t

)

Patch 2 with l=0.1

0 2000 40000

20

40

60

80

100

120

S2(t)

I 2(t

)

Patch 2 with l=1

0 2000 40000

20

40

60

80

100

120

S2(t)

I 2(t

)

Patch 2 with l=1.5

(i) (ii) (iii) (iv)

Figure 6: The phase portraits of Ii(t) vs Si(t) (i = 1, 2) for (2.15) with (4.1), ε = 1, and (i) DS1 =0, DS2 = 0, Dl1 = 0, Dl2 = 0, DI1 = 0, DI2 = 0; (ii) DS1 = 0.45, DS2 = 0.5, Dl1 = 0.03, Dl2 =0.3, DI1 = 0.09, DI2 = 0.1; (iii) DS1 = 0.45, DS2 = 0.5, Dl1 = 0.3, Dl2 = 0.3, DI1 = 0.09, DI2 =0.1; (iv) DS1 = 0.45, DS2 = 0.5, Dl1 = 0.45, Dl2 = 0.3, DI1 = 0.09, DI2 = 0.1. Initial values aregiven by (4.5).

113


0 1000 2000 30000

50

100

150

200

250

S1(t)


τ = 5

τ = 10

τ = 15

0 50000

200

400

600

800

1000

S1(t)

I 1(t

)

Patch 1 with d=0.1

τ = 5

τ = 10

τ = 15

0 2000 40000

50

100

150

200

250

300

350

400

S1(t)

I 1(t

)

Patch 1 with d=1

τ = 5

τ = 10

τ = 15

0 1000 2000 30000

20

40

60

80

100

120

140

S1(t)

I 1(t

)

Patch 1 with d=2

τ = 5

τ = 10

τ = 15

1000 2000 3000 40000

10

20

30

40

50

60

70

S2(t)

2


τ = 5

τ = 10

τ = 15

0 2000 40000

10

20

30

40

50

60

S2(t)

I 2(t

)

Patch 2 with d=0.1

τ = 5

τ = 10

τ = 15

0 2000 40000

10

20

30

40

50

60

70

80

S2(t)

I 2(t

)

Patch 2 with d=1

τ = 5

τ = 10

τ = 15

0 2000 4000 60000

20

40

60

80

100

120

S2(t)

I 2(t

)

Patch 2 with d=2

τ = 5

τ = 10

τ = 15

(i) (ii) (iii) (iv)

Figure 7: The phase portrait of Ii(t) vs Si(t)(i = 1, 2) for (2.15) with (4.5), ε = 1, Dl1 = Dl2 = DI1 =DI2 = 0 and (i) DS1 = 0, DS2 = 0; (ii) DS1 = 0.02, DS2 = 0.2; (iii) DS1 = 0.2, DS2 = 0.2; (iv)DS1 = 0.4, DS2 = 0.2.

114


0 1000 2000 30000

50

100

150

200

250

S1(t)


τ = 5

τ = 10

τ = 15

0 2000 40000

100

200

300

400

500

600

S1(t)

I 1(t

)

Patch 1 with l=0.1

τ = 5

τ = 10

τ = 15

0 2000 40000

50

100

150

200

250

300

350

400

S1(t)

I 1(t

)

Patch 1 with l=0.5

τ = 5

τ = 10

τ = 15

0 2000 40000

50

100

150

S1(t)

I 1(t

)

Patch 1 with l=1.5

τ = 5

τ = 10

τ = 15

1000 2000 3000 40000

10

20

30

40

50

60

70

S2(t)

2


τ = 5

τ = 10

τ = 15

0 2000 40000

10

20

30

40

50

60

70

80

S2(t)

I 2(t

)

Patch 2 with l=0.1

τ = 5

τ = 10

τ = 15

0 2000 40000

50

100

150

S2(t)

I 2(t

)

Patch 2 with l=0.5

τ = 5

τ = 10

τ = 15

0 2000 40000

50

100

150

S2(t)

I 2(t

)

Patch 2 with l=1.5

τ = 5

τ = 10

τ = 15

(i) (ii) (iii) (iv)

Figure 8: The phase portrait of Ii(t) vs Si(t)(i = 1, 2) with (4.5), ε = 1, and (i) DS1 = 0, DS2 =0, Dl1 = 0, Dl2 = 0, DI1 = 0, DI2 = 0; (ii) DS1 = 0.45, DS2 = 0.5, Dl1 = 0.03, Dl2 = 0.3, DI1 =0.09, DI2 = 0.1; (iii) DS1 = 0.45, DS2 = 0.5, Dl1 = 0.15, Dl2 = 0.3, DI1 = 0.09, DI2 = 0.1; (iv)DS1 = 0.45, DS2 = 0.5, Dl1 = 0.45, Dl2 = 0.3, DI1 = 0.09, DI2 = 0.1.

115


5. Conclusion and discussionIn this paper, we have derived a new epidemic model to describe the dynamics of diseases with afixed latency in a two-patch environment. Starting from the classic Kermack-McKendrick model,making use of a fundamental partial differential equation for the evolution of diseases with in-fection age and time, and tracking the dispersal of latent individuals, we have obtained a modelin the form of a system of delay differential equations which, in addition to the linear dispersionterms, contains non-local infection terms. The patches can be communities, cities, regions andeven countries; and the population dispersal among patches can be interpreted as the movementsby which people travel or migrate between patches.

For this new model, we have verified the positivity and boundedness of the solutions to thesystem. Since the model does not consider a demographic structure for the population, as in theclassical Kermack-McKendrick SIR model, we have shown that this new model also does notallow the disease to persist, meaning that the disease eventually dies out in both patches. In sucha situation, final sizes S1(∞) and S2(∞), as well as the patterns by which the disease dies out inthe two patches stand out as important issues for the model. For the former, unfortunately we cannot derive equations similar to (1.3) for the classical Kermack-McKendrick SIR model that cancompletely determine the final sizes in the two patches. However, we have proved that the ratioof the two final sizes for the new model (2.15) is totally determined by the ratio of the dispersionrates of the susceptible individuals between the two patches. An immediate implication of thisratio result is that restricting travel from Patch 1 to Patch 2 (decreasing DS1) will increase S1(∞)and hence will eventually benefit Patch 1. A similar conclusion applies to the consequence ofrestricting travel from Patch 2 to Patch 1. This very well justifies the necessity of banning of travelto those disease areas by a city or a country.

Theoretically determining the patterns by which the disease dies out for this new model doesnot seem to be possible, due to the inclusion of both the latent delay and the non-local infectionterms in the model. We have numerically studied this problem by examining the effect of the lengthof disease latency and the strengths of spatial dispersion of individuals in different classes, partic-ularly the impact of the latency and non-locality caused by the mobility of the latent individuals onthe disease dynamics. In general, the impact is complicated. We have numerically observed that(i) either the dispersion or the disease latency can cause multiple outbreaks; (ii) the number of out-breaks in the two patches can be different due to unbalanced travels; (iii) a larger latency tends tocause more outbreaks, but in the mean time, decrease the maximum outbreak sizes; (iv) relativelylarger values of DS1 (DS2) tend to raise the maximum outbreak sizes in Patch 2 (Patch 1). Thesefindings are in strong contrast to the disease dynamics of the classical Kermack-McKendrick SIRmodel (1.1). While the length of latent period is disease specific and nothing can be done to it, thedispersion rates are controllable via issuing restrictions or bans on the travels between patches. Ourmodel and the results about the model provide some insights into how to avoid multiple outbreaksand how to predict and control the maximum outbreak sizes in the two patches.

We point out that in our new model (2.15), we have omitted the demographic structure byignoring the recruitment (including births) and the natural deaths. This is reasonable for a fast dis-

116


ease, since in such a case the disease mean life time is much shorter than the mean life expectationof the population. For a disease with longer life time, this model becomes unrealistic and modi-fications are needed to include the demographic structure. In a forthcoming paper, we extend themodel (2.15) to one whose underlying population model (an equation that governs the populationgrowth in the absence of disease) supports a globally asymptotically stable positive equilibrium. Insuch a situation, the disease free equilibrium may or may not be stable, and the basic reproductionnumber is closely related to the stability of this disease free equilibrium as well as to the existenceof an endemic equilibrium. Along this line, there have been extensive works on epidemic modelson patch environments; see e.g., [2, 3, 8, 12, 14, 15, 16] and the references therein. However, to theauthors’ best knowledge, none of them considers the non-local infections caused by a fixed latenttime and the mobility of individuals during this latent period.

AcknowledgementsThis research was supported by NSERC, by NCE-MITACS of Canada and by PREA of Ontario.

References[1] R. M. Anderson, R. M. May. Infectious diseases of humans: dynamics and control, Oxford

University Press, Oxford, UK, 1991.

[2] J. Arino, P. van den Driessche. A multi-city epidemic model. Math. Popul. Stud., 10 (2003),175-193.

[3] J. Arino, P. van den Driessche. The basic reproduction number in a multi-city compartmentalepidemic model. LNCIS, 294 (2003), 135-142.

[4] F. Brauer. Some simple epidemic models. Math. Biosci. Engin., 3 (2006), 1-15.

[5] O. Diekmann, J. A. P. Heesterbeek. Mathematical epidemiology of infectious diseases:model building, analysis and interpretation. Wiley, 2000.

[6] J. K. Hale, S. M. Verduyn Lunel. Introduction to functional differential equations. Spring-Verlag, New York, 1993.

[7] W. O. Kermack, A. G. McKendrick. A contribution to the mathematical theory of epidemics.Proc. Royal Soc. London, 115 (1927), 700-721.

[8] Y.-H. Hsieh, P. van den Driessche, L. Wang. Impact of travel between patches for spatialspread of disease. Bull. Math. Biol., 69 (2007), 1355-1375.

[9] J. A. J. Metz, O. Diekmann. The dynamics of physiologically structured populations.Springer-Verlag, New York, 1986.

117


[10] K. Mischaikow, H. Smith, H. R. Thieme. Asymptotically autonomous semiflows: chain re-currence and Lyapunov functions. Trans. Amer. Math. Soc., 347 (1995), 1669-1685.

[11] J. D. Murray. Mathematical biology. 3rd ed., Springer-Verlag, New York, 2002.

[12] M. Salmani, P. van den Driessche. A model for disease transmission in a patchy environment.Disc. Cont. Dynam. Syst. Ser. B, 6 (2006), 185-202.

[13] H. R. Thieme, C. Castillo-Chavez. Asymptotically autonomous epidemic models, in Mathe-matical Population Dynamics: Analysis of Heterogeneity, Vol. 1: Theory of Epidemics (O.Arino, D. Axelrod, M. Kimmel, M. Langlais eds.), pp. 33-50, Wuerz, 1995.

[14] W. Wang, X.-Q. Zhao. An epidemic model in a patchy environment. Math. Biosci., 190(2004), 97-112.

[15] W. Wang, X.-Q. Zhao. An age-structured epidemic model in a patchy environment. SIAM J.Appl. Math., 65 (2005), 1597-1614.

[16] W. Wang, X.-Q. Zhao. An epidemic model with population dispersal and infection period.SIAM J. Appl. Math., 66 (2006), 1454-1472.

118

generalization of the kermack-mckendrick sir model to a ...generalization of the kermack-mckendrick...

Documents